In linear regression, why after normalization of both x and y, the slope becomes cor(x, y)?

Question

If we normalize (scale and center) x and y, then the slope from y_norm ~ x_norm is equal to cor(x, y). Why?

score 2 · Answer 1 · answered Oct 25 '16 at 03:13

2

Because, by definition y_norm and x_norm are unit-variance random variables whose covariance is related to the covariance of $X$ and $Y$ by $$\operatorname{cov}\left(\frac{X-\mu_X}{\sigma_X},\frac{Y-\mu_Y}{\sigma_Y}\right) = \frac{\operatorname{cov}(X,Y)}{\sigma_X\sigma_Y}.$$ Now apply the definition of cor(x,y).

answered Oct 25 '16 at 03:13

Dilip Sarwate

46,658

My question is very similar to: – Alireza Oct 25 '16 at 05:10
http://stats.stackexchange.com/questions/183778/apparent-contradiction-in-inverting-linear-regression – Alireza Oct 25 '16 at 05:10
Not sure how to do a newline here! Anyways, I'm still not sure how the slope becomes equal to the correlation. And also the answers to the above post are not very clear. I came upon this problem when I was trying both y_norm ~ x_norm and x_norm ~ y_norm, and to my surprise, the slope were the same. – Alireza Oct 25 '16 at 05:13
See this question and its answers. – Dilip Sarwate Oct 25 '16 at 16:35

score 0 · Answer 2 · answered Oct 27 '16 at 06:10

0

cor(x, y)=Cov(x,y)/Sd(x)sd(y) when you standardize.... your standard deviation of x and y will equal to 1 and therefore you will have 1 on your denominator and your cor(x,y) = cov(x,y)

answered Oct 27 '16 at 06:10

RomRom

363

I understand this, but my whole question is why cov(x, y) is the same as slope of y ~ x. – Alireza Oct 28 '16 at 22:07

In linear regression, why after normalization of both x and y, the slope becomes cor(x, y)?

2 Answers2